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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Whitehead groups of certain semidirect products of free groups


Author: Koo Guan Choo
Journal: Proc. Amer. Math. Soc. 43 (1974), 26-30
MSC: Primary 18F25
DOI: https://doi.org/10.1090/S0002-9939-1974-0338124-4
MathSciNet review: 0338124
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Abstract: Let $ D = {F_1} \times {F_2} \times \cdots \times {F_n}$ be a direct product of $ n$ free groups $ {F_1},{F_2}, \cdots ,{F_n},\alpha $ an automorphism of $ D$ which leaves all but one of the noncyclic factors in $ D$ pointwise fixed, $ T$ an infinite cyclic group and $ F$ another free group. Let $ D{ \times _\alpha }T$ be the semidirect product of $ D$ and $ T$ with respect to $ \alpha $ and $ (D{ \times _\alpha }T){ \times _{\alpha \times \text{id}T}}F$ the semidirect product of $ D{ \times _\alpha }T$ and $ F$ with respect to the automorphism $ \alpha \times idT$ of $ D{ \times _\alpha }T$ induced by $ \alpha $. We prove that the Whitehead group of $ (D{ \times _\alpha }T){ \times _{\alpha \times idT}}F$ and the projective class group of the integral group ring $ Z((D{ \times _\alpha }T){ \times _{\alpha \times idT}}F)$ are trivial. These results extend that of [3].


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DOI: https://doi.org/10.1090/S0002-9939-1974-0338124-4
Keywords: Whitehead group, projective class group, semidirect product of free groups, twisted group ring
Article copyright: © Copyright 1974 American Mathematical Society